Search results for: Fractals

Authors: Harsha Gopalakrishnan, Srijanani Anurag Prasad

Abstract:

Quadrilateral Labyrinth Fractals are a type of fractals presented in this paper. They belong to a unique class of fractals on any plane quadrilateral. The previously researched labyrinth fractals on the unit square and triangle inspire this form of fractal. This work describes how to construct a quadrilateral labyrinth fractal and looks at the circumstances in which it can be understood as the attractor of an iterated function system. Furthermore, some of its topological properties and the Hausdorff and box-counting dimensions of the quadrilateral labyrinth fractals are studied.

Keywords: Fractals, labyrinth fractals, dendrites, iterated function system, non-self similar, non-self affine, connected, path connected.

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Authors: R. Uthayakumar, D. Easwaramoorthy

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This paper investigates the fractals generated by the dynamical system of intuitionistic fuzzy contractions in the intuitionistic fuzzy metric spaces by generalizing the Hutchinson-Barnsley theory. We prove some existence and uniqueness theorems of fractals in the standard intuitionistic fuzzy metric spaces by using the intuitionistic fuzzy Banach contraction theorem. In addition to that, we analyze some results on intuitionistic fuzzy fractals in the standard intuitionistic fuzzy metric spaces with respect to the Hausdorff intuitionistic fuzzy metrics.

Keywords: Fractal Analysis, Fixed Point, Contraction, Iterated Function System, Intuitionistic Fuzzy Metric Space.

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Authors: Timothee G. Leleu

Abstract:

A new technique of topological multi-scale analysis is introduced. By performing a clustering recursively to build a hierarchy, and analyzing the co-scale and intra-scale similarities, an Iterated Function System can be extracted from any data set. The study of fractals shows that this method is efficient to extract self-similarities, and can find elegant solutions the inverse problem of building fractals. The theoretical aspects and practical implementations are discussed, together with examples of analyses of simple fractals.

Keywords: hierarchical clustering, multi-scale analysis, Similarity hashing.

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Authors: Mohammadsadegh Zanganehfar

Abstract:

Since the introduction of fractal geometry in 1970, numerous efforts have been made by architects and researchers to transfer this area of mathematical knowledge in the discipline of architecture and postmodernist discourse. The discourse of complexity and architecture is one of the most significant ongoing discourses in the discipline of architecture from the 70's until today and has generated significant styles such as deconstructivism and parametricism in architecture. During these years, several projects were designed and presented by designers and architects using fractal geometry, but due to the lack of sufficient knowledge and appropriate comprehension of the features and characteristics of this nonlinear geometry, none of the fractal-based designs have been successful and satisfying. Fractal geometry as a geometric technology has a long presence in the history of architecture. The current research attempts to identify and discover the characteristics, features, potentials and functionality of fractals despite their aesthetic aspect by examining case studies of pre-modern architecture in Asia and investigating the function of fractals. 

Keywords: Asian architecture, fractal geometry, fractal technique, geometric properties

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Authors: Vladislav N. Dumachev, Vladimir A. Rodin

Abstract:

By utilizing the system of the recurrence equations, containing two parameters, the dynamics of two antagonistically interconnected populations is studied. The following areas of the system behavior are detected: the area of the stable solutions, the area of cyclic solutions occurrence, the area of the accidental change of trajectories of solutions, and the area of chaos and fractal phenomena. The new two-dimensional diagram of the dynamics of the solutions change (the fractal cabbage) has been obtained. In the cross-section of this diagram for one of the equations the well-known Feigenbaum tree of doubling has been noted.Keywordsbifurcation, chaos, dynamics of populations, fractals

Keywords: bifurcation, chaos, dynamics of populations, fractals

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Authors: G. N. Srinivasan, G. Shobha

Abstract:

This paper presents an overview of the methodologies and algorithms for statistical texture analysis of 2D images. Methods for digital-image texture analysis are reviewed based on available literature and research work either carried out or supervised by the authors.

Keywords: Image Texture, Texture Analysis, Statistical Approaches, Structural approaches, spectral approaches, Morphological approaches, Fractals, Fourier Transforms, Gabor Filters, Wavelet transforms.

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Authors: R. Uthayakumar, G. Arockia Prabakar

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In this paper, we introduce R Iterated Function System and employ the Hutchinson Barnsley theory (HB) to construct a fractal set as its unique fixed point by using Reich contractions in a complete b metric space. We discuss about well posedness of fixed point problem for b metric space.

Keywords: Fractals, Iterated Function System, Compact set, Reich Contraction, Well posedness.

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Authors: R. Uthayakumar, D. Easwaramoorthy

Abstract:

The purpose of this paper is to present the fuzzy contraction properties of the Hutchinson-Barnsley operator on the fuzzy hyperspace with respect to the Hausdorff fuzzy metrics. Also we discuss about the relationships between the Hausdorff fuzzy metrics on the fuzzy hyperspaces. Our theorems generalize and extend some recent results related with Hutchinson-Barnsley operator in the metric spaces.

Keywords: Fractals, Iterated Function System, Hutchinson- Barnsley Operator, Fuzzy Metric Space, Hausdorff Fuzzy Metric.

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Authors: Amar Partap Singh Pharwaha, Shweta Rani

Abstract:

To develop a reliable and cost effective communication platform for the telemedicine applications, novel antenna design has been presented using bacterial foraging optimization (BFO) technique. The proposed antenna geometry is achieved by etching a modified Koch curve fractal shape at the edges and a square shape slot at the center of the radiating element of a patch antenna. It has been found that the new antenna has achieved 43.79% size reduction and better resonating characteristic than the original patch. Representative results for both simulations and numerical validations are reported in order to assess the effectiveness of the developed methodology.

Keywords: BFO, electrical permittivity, fractals, Koch curve.

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Authors: Amar Partap Singh Pharwaha, Shweta Rani

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This paper is aimed at proposing a rhombus shaped wearable fractal antenna for wireless communication systems. The geometrical descriptors of the antenna have been obtained using bacterial foraging optimization (BFO) for wide band operation. The method of moment based IE3D software has been used to simulate the antenna and observed that miniaturization of 13.08% has been achieved without degrading the resonating properties of the proposed antenna. An analysis with different substrates has also been done in order to evaluate the effectiveness of electrical permittivity on the presented structure. The proposed antenna has low profile, light weight and has successfully demonstrated wideband and multiband characteristics for wearable electronic applications.

Keywords: BFO, bandwidth, electrical permittivity, fractals, wearable antenna.

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Authors: Truong Quang Dang Khoa, Masahiro Nakagawa

Abstract:

Chaos and fractals are novel fields of physics and mathematics showing up a new way of universe viewpoint and creating many ideas to solve several present problems. In this paper, a novel algorithm based on the chaotic sequence generator with the highest ability to adapt and reach the global optima is proposed. The adaptive ability of proposal algorithm is flexible in 2 steps. The first one is a breadth-first search and the second one is a depth-first search. The proposal algorithm is examined by 2 functions, the Camel function and the Schaffer function. Furthermore, the proposal algorithm is applied to optimize training Multilayer Neural Networks.

Keywords: learning and evolutionary computing, Chaos Optimization Algorithm, Artificial Neural Networks, nonlinear optimization, intelligent computational technologies.

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Authors: Mohammadsadegh Zanganehfar

Abstract:

Through the emergence of modern architecture, an aggressive desire for new design theories appeared through the works of architects and critics. The discourse of complexity and volumetric composition happened to be an important and controversial issue in the discipline of architecture which was discussed through a general point of view in Robert Venturi and Denise Scott Brown's book “Complexity and contradiction in architecture” in 1966, this paper attempts to identify chaos theory as a scientific model of complexity and its relation to architecture design theory by conducting a qualitative analysis and multidisciplinary critical approach through architecture and basic sciences resources. Accordingly, we identify chaotic architecture as the correlation between chaos theory and the discipline of architecture, and as an independent nonlinear design theory with specific characteristics and properties.

Keywords: Architecture complexity, chaos theory, fractals, nonlinear dynamic systems, nonlinear ontology.

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Authors: Khaled Harrar, Rachid Jennane

Abstract:

The purpose of this study is the discrimination of 28 postmenopausal with osteoporotic femoral fractures from an agematched control group of 28 women using texture analysis based on fractals. Two pre-processing approaches are applied on radiographic images; these techniques are compared to highlight the choice of the pre-processing method. Furthermore, the values of the fractal dimension are compared to those of the fractal signature in terms of the classification of the two populations. In a second analysis, the BMD measure at proximal femur was compared to the fractal analysis, the latter, which is a non-invasive technique, allowed a better discrimination; the results confirm that the fractal analysis of texture on calcaneus radiographs is able to discriminate osteoporotic patients with femoral fracture from controls. This discrimination was efficient compared to that obtained by BMD alone. It was also present in comparing subgroups with overlapping values of BMD.

Keywords: Osteoporosis, fractal dimension, fractal signature, bone mineral density.

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Authors: Nadia M. G. AL-Sa'idi, Muhammad Rushdan Md. Sd., Adil M. Ahmed

Abstract:

The IFS is a scheme for describing and manipulating complex fractal attractors using simple mathematical models. More precisely, the most popular “fractal –based" algorithms for both representation and compression of computer images have involved some implementation of the method of Iterated Function Systems (IFS) on complete metric spaces. In this paper a new generalized space called Multi-Fuzzy Fractal Space was constructed. On these spases a distance function is defined, and its completeness is proved. The completeness property of this space ensures the existence of a fixed-point theorem for the family of continuous mappings. This theorem is the fundamental result on which the IFS methods are based and the fractals are built. The defined mappings are proved to satisfy some generalizations of the contraction condition.

Keywords: Fuzzy metric space, Fuzzy fractal space, Multi fuzzy fractal space.

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Authors: A. Abou El-Azm Aly, A. Chong, F. Nicolleau, S. Beck

Abstract:

The fractal-shaped orifices are assumed to have a significant effect on the pressure drop downstream pipe flow due to their edge self-similarity shape which enhances the mixing properties. Here, we investigate the pressure drop after these fractals using a digital micro-manometer at different stations downstream a turbulent flow pipe then a direct comparison has been made with the pressure drop measured from regular orifices with the same flow area. Our results showed that the fractal-shaped orifices have a significant effect on the pressure drop downstream the flow. Also the pressure drop measured across the fractal-shaped orifices is noticed to be lower that that from ordinary orifices of the same flow areas. This result could be important in designing piping systems from point of view of losses consideration with the same flow control area. This is promising to use the fractal-shaped orifices as flowmeters as they can sense the pressure drop across them accurately with minimum losses than the regular ones.

Keywords: Fractal-shaped orifice, pressure drop, turbulent flow.

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Authors: I. Yakubu, Y. Y. Ziggah, C. Yeboah

Abstract:

In the mining environment, tailings dam embankment is among the hazards and risk areas. The tailings dam embankment could fail and result to damages to facilities, human injuries or even fatalities. Periodic monitoring of the dam embankment is needed to help assess the safety of the tailings dam embankment. Artificial intelligence techniques such as fractals can be used to analyse the stability of the monitored dataset from survey measurement techniques. In this paper, the fractal dimension (D) was determined using D = 2-H. The Hurst parameters (H) of each monitored prism were determined by using a time domain of rescaled range programming in MATLAB software. The fractal dimensions of each monitored prism were determined based on the values of H. The results reveal that the values of the determined H were all within the threshold of 0 ≤ H ≤ 1 m. The smaller the H, the bigger the fractal dimension is. Fractal dimension values ranging from 1.359 x 10^-4 m to 1.8843 x 10^-3 m were obtained from the monitored prisms on the based on the tailing dam embankment dataset used. The ranges of values obtained indicate that the tailings dam embankment is stable.

Keywords: Hurst parameter, fractal dimension, tailings dam embankment, surveyed dataset.

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Authors: Y. Laaroussi, Z. Guennoun, A. Amar

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Modeling and forecasting dynamics of rainfall occurrences constitute one of the major topics, which have been largely treated by statisticians, hydrologists, climatologists and many other groups of scientists. In the same issue, we propose, in the present paper, a new hybrid method, which combines Extreme Values and fractal theories. We illustrate the use of our methodology for transformed Emberger Index series, constructed basing on data recorded in Oujda (Morocco). The index is treated at first by Peaks Over Threshold (POT) approach, to identify excess observations over an optimal threshold u. In the second step, we consider the resulting excess as a fractal object included in one dimensional space of time. We identify fractal dimension by the box counting. We discuss the prospect descriptions of rainfall data sets under Generalized Pareto Distribution, assured by Extreme Values Theory (EVT). We show that, despite of the appropriateness of return periods given by POT approach, the introduction of fractal dimension provides accurate interpretation results, which can ameliorate apprehension of rainfall occurrences.

Keywords: Extreme values theory, Fractals dimensions, Peaks Over Threshold, Rainfall occurrences.

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Authors: L. Bennaceur Farah, I. R. Farah, R. Bennaceur, Z. Belhadj, M. R. Boussema

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The overall objective of this paper is to retrieve soil surfaces parameters namely, roughness and soil moisture related to the dielectric constant by inverting the radar backscattered signal from natural soil surfaces. Because the classical description of roughness using statistical parameters like the correlation length doesn't lead to satisfactory results to predict radar backscattering, we used a multi-scale roughness description using the wavelet transform and the Mallat algorithm. In this description, the surface is considered as a superposition of a finite number of one-dimensional Gaussian processes each having a spatial scale. A second step in this study consisted in adapting a direct model simulating radar backscattering namely the small perturbation model to this multi-scale surface description. We investigated the impact of this description on radar backscattering through a sensitivity analysis of backscattering coefficient to the multi-scale roughness parameters. To perform the inversion of the small perturbation multi-scale scattering model (MLS SPM) we used a multi-layer neural network architecture trained by backpropagation learning rule. The inversion leads to satisfactory results with a relative uncertainty of 8%.

Keywords: Remote sensing, rough surfaces, inverse problems, SAR, radar scattering, Neural networks and Fractals.

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Authors: Shimal Das, Dibyendu Ghoshal

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Fractal based digital image compression is a specific technique in the field of color image. The method is best suited for irregular shape of image like snow bobs, clouds, flame of fire; tree leaves images, depending on the fact that parts of an image often resemble with other parts of the same image. This technique has drawn much attention in recent years because of very high compression ratio that can be achieved. Hybrid scheme incorporating fractal compression and speedup techniques have achieved high compression ratio compared to pure fractal compression. Fractal image compression is a lossy compression method in which selfsimilarity nature of an image is used. This technique provides high compression ratio, less encoding time and fart decoding process. In this paper, fractal compression with quad tree and DCT is proposed to compress the color image. The proposed hybrid schemes require four phases to compress the color image. First: the image is segmented and Discrete Cosine Transform is applied to each block of the segmented image. Second: the block values are scanned in a zigzag manner to prevent zero co-efficient. Third: the resulting image is partitioned as fractals by quadtree approach. Fourth: the image is compressed using Run length encoding technique.

Keywords: Fractal coding, Discrete Cosine Transform, Iterated Function System (IFS), Affine Transformation, Run length encoding.

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Authors: N.E.Galich

Abstract:

We develop new nonlinear methods of immunofluorescence analysis for a sensitive technology of respiratory burst reaction of DNA fluorescence due to oxidative activity in the peripheral blood neutrophils. Histograms in flow cytometry experiments represent a fluorescence flashes frequency as functions of fluorescence intensity. We used the Shannon-Weaver index for definition of neutrophils- biodiversity and Hurst index for definition of fractal-s correlations in immunofluorescence for different donors, as the basic quantitative criteria for medical diagnostics of health status. We analyze frequencies of flashes, information, Shannon entropies and their fractals in immunofluorescence networks due to reduction of histogram range. We found the number of simplest universal correlations for biodiversity, information and Hurst index in diagnostics and classification of pathologies for wide spectra of diseases. In addition is determined the clear criterion of a common immunity and human health status in a form of yes/no answers type. These answers based on peculiarities of information in immunofluorescence networks and biodiversity of neutrophils. Experimental data analysis has shown the existence of homeostasis for information entropy in oxidative activity of DNA in neutrophil nuclei for all donors.

Keywords: blood and cells fluorescence in diagnostics ofdiseases, cytometric histograms, entropy and information in fractalnetworks of oxidative activity of DNA, long-range chromosomalcorrelations in living cells.

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Authors: Srinivasan Aruchamy, Partha Bhattacharjee, Goutam Sanyal

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Optic disk segmentation plays a key role in the mass screening of individuals with diabetic retinopathy and glaucoma ailments. An efficient hardware-based algorithm for optic disk localization and segmentation would aid for developing an automated retinal image analysis system for real time applications. Herein, TMS320C6416DSK DSP board pixel intensity based fractal analysis algorithm for an automatic localization and segmentation of the optic disk is reported. The experiment has been performed on color and fluorescent angiography retinal fundus images. Initially, the images were pre-processed to reduce the noise and enhance the quality. The retinal vascular tree of the image was then extracted using canny edge detection technique. Finally, a pixel intensity based fractal analysis is performed to segment the optic disk by tracing the origin of the vascular tree. The proposed method is examined on three publicly available data sets of the retinal image and also with the data set obtained from an eye clinic. The average accuracy achieved is 96.2%. To the best of the knowledge, this is the first work reporting the use of TMS320C6416DSK DSP board and pixel intensity based fractal analysis algorithm for an automatic localization and segmentation of the optic disk. This will pave the way for developing devices for detection of retinal diseases in the future.

Keywords: Color retinal fundus images, Diabetic retinopathy, Fluorescein angiography retinal fundus images, Fractal analysis.

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